An Exact Dominant Degree Condition for Transitive Tournament Factors in Digraphs

Abstract

Let $r\ge 2$, let $T_r$ denote the transitive tournament on $r$ vertices, and write $d_G^{\ast }(v):=\max \{d_G^{+}(v),d_G^{-}(v)\}$. We prove that if $r\mid n$ and an $n$-vertex digraph $G$ satisfies $d_G^{\ast }(x)+d_G^{\ast }(y)\ge 2(1-1/r)n-1$ for every $x\ne y\in V(G)$ with $xy\notin E(G)$, then $G$ has a $T_r$-factor, and the bound is best possible. Furthermore, by applying our main theorem, we settle Treglown's conjecture on the dominant degree $d_G^{\ast }(x)\ge (1-1/r)n$ and answer Molla and Treglown's problem of determining the exact Ore-type threshold $2(1-1/r)n-1$, and we obtain stronger versions of the theorems of Czygrinow, DeBiasio, Kierstead and Molla.

The main theorem is wrong